I assume it simply means that the sacrifice that takes place at Mass is not bloody. There is blood on the Altar yes but not in the way we see in the Old Testament temple where the blood of dead animals was splashed up the sides of the Altar and burned on top of it.

Also compared to the sacrifice as it first took place on the cross the manner is un-bloody. Our Lord on the cross was pouring blood from His hands, feet, head and side not to mention His back and legs where He has been scourged. Instead of these bloody sacrifices God has made the sacrafice for us once and for all in a bloody manner and we now partake of that same sacrafice each Mass in an un-bloody way.

The Mass can’t be called bloodless, because the blood of Christ really is offered in it. When the Council of Trent said the Mass is an “unbloody” sacrifice, it meant Jesus isn’t slain again, not that his blood isn’t present (which would have denied transubstantiation).

[Christ], our Lord and God, was once and for all to offer himself to God the Father by his death on the altar of the cross, to accomplish there an everlasting redemption. But because his priesthood was not to end with his death, at the Last Supper “on the night when he was betrayed,” [he wanted] to leave to his beloved spouse the Church a visible sacrifice (as the nature of man demands) by which the bloody sacrifice which he was to accomplish once for all on the cross would be re-presented, its memory perpetuated until the end of the world, and its salutary power be applied to the forgiveness of the sins we daily commit.

“Different manner” and “Unbloody” seems to imply that he is not making a new sacrifice or doing it over again, but that the manner of the sacrifice of the mass is that it is “making present” the once and for all sacrifice on the cross.

As I understand it, when something is offered to God in sacrifice, it can be offered in various ways. For instance, a living thing (such as a lamb) can be killed and entirely consumed by fire (a holocaust) or it can be killed and eaten by those offering the sacrifice (like the passover lamb) or it is not killed but simply presented alive to or “waved” before God (as when the entire tribe of Levi was sacrificed as a wave offering). Sacrifices where a living victim is killed are called “bloody”; those where the victim is not killed are called “unbloody.” In the sacrifice of the Mass, the risen Lord Jesus Christ is not killed or molested in any way but simply presented to or “waved” before God. Since the wounds of his crucifixion are still present in his risen body, simply presenting him to God is a re-presentation of his sacrifice on the cross, as if it say, See the wounds in his hands, feet and side and, for the sake of his sorrowful passion, have mercy on us and on the whole world.